Sums of characteristic values of compact polynomial operator pencils

We consider polynomial pencils whose coefficients are compact operators. Bounds for the sums of absolute values, real and imaginary parts of characteristic values are derived. Applications to differential and difference equations are discussed.     

Trace formula for polynomial operator pencils and its applications

Mathematical Institute and Computing Center, Academy of Sciences of the Tadzhik SSR. M. V. Lomonosov Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 4, pp. 90–92, October–December, 1991.     

Banach algebras associated with linear operator pencils

A direct relationship between the theory of pseudoresolvents and the spectral theory of linear operator pencils is established.     

On operator factorization of linear relations

Assume that $A$ and $B$ are two linear relations in certain linear spaces. The main objective of this note is to present two characterizations of the existence of two linear operators $C$ and $T$ such that $A = BC$ and $A = T\!B$ , respectively. These factorizations extend and complete some known results due to R.G. Douglas, Z. Sebestyén and D. Popovici.     

Remainder estimates in spectral asymptotic formulas for linear operator pencils

One gives a qualified remainder estimate in the asymptotic formulas of the distribution functions of the positive and negative spectra of a linear pencil of differential operators, where the lowest operator is not assumed to be elliptic.Translated from Problemy Maternaticheskogo Analiza, No. 9, pp. 34–56, 1984.     

On spectra of a certain class of quadratic operator pencils with one-dimensional linear part

We consider a class of quadratic operator pencils that occur in many problems of physics. The part of such a pencil linear with respect to the spectral parameter describes viscous friction in problems of small vibrations of strings and beams. Patterns in the location of eigenvalues of such pencils are established. If viscous friction (damping) is pointwise, then the operator in the linear part of the pencil is one-dimensional. For this case, rules in the location of purely imaginary eigenvalues are found. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 5, pp. 702–716, May, 2007.     

Sparse bivariate polynomial factorization

Motivated by Sasaki’s work on the extended Hensel construction for solving multivariate algebraic equations, we present a generalized Hensel lifting, which takes advantage of sparsity, for factoring bivariate polynomial over the rational number field. Another feature of the factorization algorithm presented in this article is a new recombination method, which can solve the extraneous factor problem before lifting based on numerical linear algebra. Both theoretical analysis and experimental data show that the algorithm is efficient, especially for sparse bivariate polynomials.     

Hermitian pencils with a cubic minimal polynomial

Let A be an n × n complex matrix and write A = H + iK, where H and K are Hermitian matrices. We show that if the minimal polynomial of the pencil xH + yK has degree 3, then there is a unitary matrix U such that U^-1AU is block diagonal with blocks of size 3 × 3 or smaller. This is a special case of a conjecture made by Kippenhahn in 1951.     

Weak stability of factorizations of operator pencils

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 51, pp. 141–143, 1989.     

Self-adjoint quadratic operator pencils and elliptic problems

Functional Analysis and Its Applications -     

Inertia theorems for operator pencils and applications

In this paper we obtain general infinite dimensional inertia theorems for linear pencils in Hilbert space which cover previously known results for the finite dimensional case and for block weighted shifts. Connections with definite subspaces for contractions in spaces with indefinite metric are discussed.